Bottom Line:
In addition, we perform a detailed and rigorous theoretical analysis of the sensitivity of the hierarchical levels to weight perturbations.In doing so, we show that the ordering of the reticular edges is more robust to noise in weight estimation than is the ordering of the tree edges.We discuss applications of this generalized Horton-Strahler ordering to the study of leaf venation and other biological networks.

Affiliation: Department of Mathematics, Duke University, Durham, North Carolina, United States of America. yury@math.duke.edu

ABSTRACTThe structure of hierarchical networks in biological and physical systems has long been characterized using the Horton-Strahler ordering scheme. The scheme assigns an integer order to each edge in the network based on the topology of branching such that the order increases from distal parts of the network (e.g., mountain streams or capillaries) to the "root" of the network (e.g., the river outlet or the aorta). However, Horton-Strahler ordering cannot be applied to networks with loops because they they create a contradiction in the edge ordering in terms of which edge precedes another in the hierarchy. Here, we present a generalization of the Horton-Strahler order to weighted planar reticular networks, where weights are assumed to correlate with the importance of network edges, e.g., weights estimated from edge widths may correlate to flow capacity. Our method assigns hierarchical levels not only to edges of the network, but also to its loops, and classifies the edges into reticular edges, which are responsible for loop formation, and tree edges. In addition, we perform a detailed and rigorous theoretical analysis of the sensitivity of the hierarchical levels to weight perturbations. In doing so, we show that the ordering of the reticular edges is more robust to noise in weight estimation than is the ordering of the tree edges. We discuss applications of this generalized Horton-Strahler ordering to the study of leaf venation and other biological networks.

pone-0036715-g002: An illustration of the loop merging procedure.The merging is applied to the graph from Fig. 1C. Red, dashed edges are the ones removed during merging, the corresponding numbers show their levels. Levels of faces is encoded by the color: white faces have level 1, light blue faces have level 2, and gold faces have level 3. Note that is the unbounded face.

Mentions:
The procedure to order planar weighted graphs can be summarized as follows. First, an order is assigned to all faces in the graph. We then iterate through edges in order of increasing weight. When a given edge is on the boundary of two distinct faces, then this edge is removed, creating a merged face. The order of this merged face follows the Horton-Strahler rule (see Eq. (1)) given the orders of the two faces. Similarly, the order of the edge to be removed is set equal to the minimum of the order of the two merged faces. A step-by-step illustration of loop merging applied to the tree in Fig. 1C is shown in Fig. 2. Notice that this procedure builds a rooted binary tree, where leaves correspond to the faces of G, and the rest of the vertices correspond to unions of these faces. The assignment of levels in this tree follows the original Horton-Strahler algorithm. It is also useful to remember that faces of G are vertices of its dual graph, , and merging faces of G can be thought of as adding an edge to . Hence, the two merging procedures that we described are, in some sense, dual. We shall refer to the binary tree of faces as the co-tree of G, and denote it by .

pone-0036715-g002: An illustration of the loop merging procedure.The merging is applied to the graph from Fig. 1C. Red, dashed edges are the ones removed during merging, the corresponding numbers show their levels. Levels of faces is encoded by the color: white faces have level 1, light blue faces have level 2, and gold faces have level 3. Note that is the unbounded face.

Mentions:
The procedure to order planar weighted graphs can be summarized as follows. First, an order is assigned to all faces in the graph. We then iterate through edges in order of increasing weight. When a given edge is on the boundary of two distinct faces, then this edge is removed, creating a merged face. The order of this merged face follows the Horton-Strahler rule (see Eq. (1)) given the orders of the two faces. Similarly, the order of the edge to be removed is set equal to the minimum of the order of the two merged faces. A step-by-step illustration of loop merging applied to the tree in Fig. 1C is shown in Fig. 2. Notice that this procedure builds a rooted binary tree, where leaves correspond to the faces of G, and the rest of the vertices correspond to unions of these faces. The assignment of levels in this tree follows the original Horton-Strahler algorithm. It is also useful to remember that faces of G are vertices of its dual graph, , and merging faces of G can be thought of as adding an edge to . Hence, the two merging procedures that we described are, in some sense, dual. We shall refer to the binary tree of faces as the co-tree of G, and denote it by .

Bottom Line:
In addition, we perform a detailed and rigorous theoretical analysis of the sensitivity of the hierarchical levels to weight perturbations.In doing so, we show that the ordering of the reticular edges is more robust to noise in weight estimation than is the ordering of the tree edges.We discuss applications of this generalized Horton-Strahler ordering to the study of leaf venation and other biological networks.

Affiliation:
Department of Mathematics, Duke University, Durham, North Carolina, United States of America. yury@math.duke.edu

ABSTRACTThe structure of hierarchical networks in biological and physical systems has long been characterized using the Horton-Strahler ordering scheme. The scheme assigns an integer order to each edge in the network based on the topology of branching such that the order increases from distal parts of the network (e.g., mountain streams or capillaries) to the "root" of the network (e.g., the river outlet or the aorta). However, Horton-Strahler ordering cannot be applied to networks with loops because they they create a contradiction in the edge ordering in terms of which edge precedes another in the hierarchy. Here, we present a generalization of the Horton-Strahler order to weighted planar reticular networks, where weights are assumed to correlate with the importance of network edges, e.g., weights estimated from edge widths may correlate to flow capacity. Our method assigns hierarchical levels not only to edges of the network, but also to its loops, and classifies the edges into reticular edges, which are responsible for loop formation, and tree edges. In addition, we perform a detailed and rigorous theoretical analysis of the sensitivity of the hierarchical levels to weight perturbations. In doing so, we show that the ordering of the reticular edges is more robust to noise in weight estimation than is the ordering of the tree edges. We discuss applications of this generalized Horton-Strahler ordering to the study of leaf venation and other biological networks.